Suppose n identical firms are racing to introduce the same innovation. The per-firm (fixed) cost of R&D is $100. R&D occurs at time t = 0. Assume for simplicity that if the innovation is developed at t = 0, then society benefits in all later periods; but if the innovation is not developed at t = 0, then research does not take place in subsequent periods. If one firm invests in R&D, it will succeed with 40% probability. All successful firms have equal chance of obtaining the patent. Assume for simplicity that the patent never expires. The discount rate is 10%

(a) (3 points) A patent owner earns a monopolistic return on the R&D cost of her/his innovation. Assume that the demand for the innovation in each period is P = 21 ? Q, where P is the price and Q is the number of units sold. The marginal cost of production equals 1. The fixed cost of production is zero. What is the monopoly profit per period from receiving the patent?

(b) (2 points) What is the expected present value of the patent when one firm invests in R&D?

(c) (3 points) What is the expected present value of the patent when two firms invest in R&D?

(d) (2 points) What is the social value of the innovation per period? Hint: Assume competitive supply (i.e., P = MC).

(e) (2 points) What is the expected discounted social value over lifetime when one firm invests in R&D? What is the expected discounted social value over lifetime when two firms invest in R&D?

(f) (7 points) What is the socially optimal number of firms investing in R&D? How many firms will invest in R&D? Please explain. To support your answers, complete the table below. That is for each n from 1 to 11, find the values of

• Probability that at least one firm will succeed, ?(n)
• Social value (expected discounted), S
• Social cost
• Net social value
• Marginal value
• Marginal cost
• Patent value (the expected present value of the patent), ?

# of Prob. Social Social Net social Marg. Marg. Patent firms value cost value value cost value 10 11 12